Automated Theorem Proving with Spider Diagrams

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Automated Theorem Proving with Spider Diagrams

• Jean Flower and Gem Stapleton
• Visual Modelling Group
• University of Brighton, UK.
• http//www.cmis.brighton.ac.uk/research/vmg

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Overview

• About reasoning with diagrams
• What is our diagrammatic notation
• What does it mean
• Reasoning
• About automated proof-writing

About reasoning with diagrams
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About reasoning with diagrams

• Euler (1700s)
• Venn (1880s)
• Shin (1995) formalised Venn DR
• Hammer (1995) formalised Euler DR
• SwobodaAllwein (current) Euler DR v2
• Implementation application in educational
  context
• Kent (1997..) spider diagrams, constraint
  diagrams
• metamodelling applications

Diagrams, HCC, VLC, InfVis, AVI, IJCAR,
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Overview

• About reasoning with diagrams
• What is our diagrammatic notation
• What does it mean
• Reasoning
• About automated proof-writing

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Concrete syntax
But how do we capture the commonality between
diagrams? What is the essential semantic
structure of a diagram? (abstract)
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Abstract syntax some language
Contour label
A
B
Shaded zone
Zone (A,B)
Spider (A,B),(A,B,)
Region set of zones
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Abstract syntax
Contour label
A
B
Shaded zone
Zone (A,B)
Spider (A,B),(A,B,)
Region set of zones
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Abstract syntax
Contour label
A
B
Shaded zone
Zone (A,B)
Spider (A,B),(A,B,)
Region set of zones
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Abstract syntax
Contour label
A
B
Shaded zone
Zone (A,B)
Spider (A,B),(A,B,)
Region set of zones
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Abstract syntax
Contour label
A
B
Shaded zone
Zone (A,B)
Spider (A,B),(A,B,)
Region set of zones
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Overview

• About reasoning with diagrams
• What is our diagrammatic notation
• What does it mean
• Reasoning
• About automated proof-writing

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Informal Semantics
A
B
Theres an element in A and everything thats in
B is also in A
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Informal Semantics
B
A
There are exactly two elements in B and
everything thats in A is also in B
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More Formally

• A isnt a set, but users interpret A as a set
• An interpretation is a pair, (U,Psi), where
• U is a set
• Psi maps contour labels to subsets of U.
• This assignment of sets to contours is enough to
  assign sets also to zones and regions

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More Formally

• Which interpretations conform to the diagram?
• An interpretation is a model for a diagram if
• Psi(r)gt number of spiders in r
• when r is shaded,
• Psi(r)lt number of spiders with a node in r
• the region consisting of all the zones maps to
  U.

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Semantics
U1,2,3, Psi(A)1, Psi(B)1,2 is a model
B
A
U1,2,3, Psi(A), Psi(B)2 is a not model
The semantics of a diagram the class of models
for that diagram
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Compound diagrams

• Increase expressiveness

Connectives. and or
U
C
B
A
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What does it mean

• Theorem
• The spider diagram language is equivalent in
  expressive power to monadic first order logic
  with equality.

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Overview

• About reasoning with diagrams
• What is our diagrammatic notation
• What does it mean
• Reasoning
• About automated proof-writing

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Reasoning

• Devise reasoning rules syntactically as
  transformations of abstract syntax
• so rules turn one abstract diagram into another
• why not transformations of concrete syntax?
• Well-formedness problems
• Need a rule to transform between equal diagrams
• consequences of working at abstract level in
  terms of presenting results back to user

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Reasoning

• Avoid interpreting in another setting (FOPL)
• because we want to allow the users to see the
  proof without a change in notation
• helps people strengthen their understanding of
  the diagammatic notation

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Reasoning
A
B
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Reasoning
Delete shading
A
B
A
B
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Reasoning
Delete shading
A
B
A
Delete a spider
B
A
B
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Reasoning

• Excluded middle

A
B
A
B
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Reasoning

• Split Spider

A
B
A
B
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Reasoning

• Combining

A
B
A
B
A
B
On a-diagrams with matching zone-sets
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Reasoning

• Combining

A
B
A
B
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Reasoning

• Other rules
• add contour
• add shaded zone
• logic rules distributivity etc
• System has been proven to be sound and complete

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Overview

• About reasoning with diagrams
• What is our diagrammatic notation
• What does it mean
• Reasoning
• About automated proof-writing

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About automated proof-writing

• Development of the proof-writer
• Unitary a-diagram ? Unitary a-diagram
• With same contours, zone sets
• Use erase shading and delete spider
• Unitary a-diagram ? disjunction a-diagram
• Find a target component in the conclusion
  diagram
• Disjunction a-diagram ? disjunction a-diagram
• For each premis component, find a target
• Turn into disjunctions of a-diagrams

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An example
A
B
A
B
D1
D2
Add Contours
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An example
A
B
A
B
D1
D2
Add Contours
A
B
A
A
B
B
D1
D2
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An example
A
B
A
B
A
B
D1
D2
Add Zones
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An example
A
B
A
B
A
B
D1
D2
Add Zones
A
B
A
A
B
B
D1
D2
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An example
A
B
A
B
A
B
D1
D2
Split Spiders
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An example
A
B
A
B
A
B
D1
D2
Split Spiders
A
B
A
B
A
B
D2
D1
A
B
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An example
A
B
A
B
A
B
D2
D1
To disjunctive normal form
A
B
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An example
A
B
A
B
A
B
D2
D1
To disjunctive normal form
A
B
A
B
D2
B
A
B
A
A
A
B
B
D1
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An example
A
B
D1
A
B
A
B
A
A
B
B
D2
Combine
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An example
A
B
D1
A
B
A
B
A
A
B
B
D2
Combine
A
B
A
B
D2
D1
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An example
A
B
A
B
D2
D1
Change D1 into D2
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An example
A
B
A
B
D2
D1
Change D1 into D2
remove false
A
B
D1
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An example
A
B
A
B
D2
D1
Change D1 into D2
delete shading
A
B
Not just deleting shading spiders
D1
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The tool

• Uses underlying metamodel of abstract syntax
• Checks which rules are applicable
• Allows users to write proofs
• Automates proofs for users
• And now a demo

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More work on proof-writing

• Which derived rules make for usable proofs?
• Heuristic approaches
• Introduce negation
• doesnt increase expressiveness
• Introduce further syntax that allows universal
  quantification and relational navigation
  (constraint diagrams).

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The end

• Any questions?
• http//www.cmis.brighton.ac.uk/research/vmg

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About automated proof-writing

• Sketch of the strategy to write proofs
• Take D1 and D2
• Remove all the ands from both sides
• Shrink D2 using excluded middle
• Add spiders and shading to D1 using excluded
  middle
• Change D1 into D2, or generate a counterexample.